Accumulated error of the uncalibrated augmented checks

PDF p. 16: Remark: uncalibrated lifts, final sentence; final S4 paragraph

For either the spin-five Æ code or the spin-four Aydin–Barg code, with initial spin $J_0=5$ or $4$, respectively, consider the proposed ideal augmented check. The code is the two-dimensional subspace $\mathcal{C}$ spanned by its displayed logical codewords; $J_z$ is the spin component whose magnetic-basis eigenvalue is $m$. The check distinguishes $\mathcal{C}$ from $\mathcal{D}=J_z\mathcal{C}$ in the initial spin manifold and applies the specified inverse isometry $W:\mathcal{D}\to \mathcal{C}$ on the $\mathcal{D}$ outcome. The lift is uncalibrated: its strength is not supplied to the recovery rule.

Adopt the general bounded diagonal lift postulate on the finite integer-spin blocks $J=0,\dots ,N$ with $N\ge J_0$, including its physical, spectral and choice hypotheses. Specialize its lift to $\mathrm{\Delta }(J,m)={\mathrm{\Delta }}_{0}Jm$ for real ${\mathrm{\Delta }}_0$. The noise evolution over an interval $\tau \ge 0$ is the exact finite GKSL channel specified there by the Hamiltonian $H$ and the three downward jumps $L_{\delta }$, $\delta =-1,0,1$. For short-checking asymptotics use its regime $\gamma \tau \ll 1$ and $\mathrm{\mid }{\mathrm{\Delta }}_0\mathrm{\mid }{J}_0^2\tau \ll 1$. The host-manifold null-branch rotation is $e^{-i\theta J_z}$ with $\theta ={\mathrm{\Delta }}_0{J}_0\tau$.

Determine the higher-order residual errors and their accumulated effect under the complete repeated-check channel, including the decay and other outcome branches of the modeled evolution. In particular, combine this noise evolution with specified outcome maps, conditional recovery operations and treatment of all outcomes sufficiently to define that channel, and determine its dependence on the lift and checking interval. The first-order isolated-rotation result below does not by itself determine this repeated-channel performance.

General bounded diagonal lift for fast-checking analysis

Proposed augmented Jz syndrome check for an uncalibrated lift

Well-defined augmented syndrome and first-order rotation correction

The article identifies the accumulated higher-order performance of this uncalibrated extension as future work. The supplied article is dated September 24, 2026. No later status search or solution is claimed here.